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In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
A clique, C, in an undirected graph G = (V, E) is a subset of the vertices, C ⊆ V, such that every two distinct vertices are adjacent. This is equivalent to the condition that the induced subgraph of G induced by C is a complete graph. In some cases, the term clique may also refer to the subgraph directly.
An undirected graph is formed by a finite set of vertices and a set of unordered pairs of vertices, which are called edges. By convention, in algorithm analysis, the number of vertices in the graph is denoted by n and the number of edges is denoted by m. A clique in a graph G is a complete subgraph of G.
The number of different Hamiltonian cycles in a complete undirected graph on n vertices is (n – 1)! / 2 and in a complete directed graph on n vertices is (n – 1)!. These counts assume that cycles that are the same apart from their starting point are not counted separately.
A graph is planar if it contains as a subdivision neither the complete bipartite graph K 3,3 nor the complete graph K 5. Another problem in subdivision containment is the Kelmans–Seymour conjecture: Every 5-vertex-connected graph that is not planar contains a subdivision of the 5-vertex complete graph K 5.
The matching complex of a complete bipartite graph K m,n is known as a chessboard complex. It is the clique graph of the complement graph of a rook's graph, [5] and each of its simplices represents a placement of rooks on an m × n chess board such that no two of the rooks attack each other.
Walecki's Hamiltonian decomposition of the complete graph . In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles. Hamiltonian decompositions have been studied both for undirected graphs and for directed graphs.